Self-intersection detection and elimination in freeform curves and surfaces

We present several algorithms for self-intersection detection, and possible elimination, in freeform planar curves and surfaces. Both local and global self-intersections are eliminated using a binormal-line criterion and a simple direct algebraic elimination procedure that enables the direct solution of the algebraic (self-)intersection constraints.All algorithms have been fully implemented and tested. Examples are presented for applications in self-intersection detection, self-intersection-free metamorphosis of curves, and proper trimming of self-intersections in offset curves.     

Canal surfaces with rational contour curves and blends bypassing the obstacles

In this paper, we will present an algebraic condition, see (20), which guarantees that a canal surface, given by its rational medial axis transform (MAT), possesses rational generalized contours (i.e., contour curves with respect to a given viewpoint). The remaining computational problem of this approach is how to find the right viewpoint. The canal surfaces fulfilling this distinguished property are suitable for being taken as modeling primitives when some rational approximations of canal surfaces are required. Mainly, we will focus on the low-degree cases such as quadratic and cubic MATs that are especially useful for applications. To document a practical usefulness of the presented approach, we designed and implemented two simple algorithms for computing rational offset blends between two canal surfaces based on the contour method which do not need any further advanced formalism (as e.g. interpolations with MPH curves). A main advantage of the designed blending technique is its simplicity and also an adaptivity to choose a suitable blend satisfying certain constrains (avoiding obstacles, bypassing other objects, etc.). Compared to other similar methods, our approach requires only one SOS decomposition for the whole family of rational canal surfaces sharing the same silhouette, which significantly simplifies the computational complexity.     

平面代数曲线间最近距离的计算

通过几何观察,指出一条曲线上的最近点是另一条曲线的等距曲线与该曲线的切点这一事实,同时提出基于等距思想的方法来求解2条平面代数曲线间的最近距离．该方法几何意义明显,可同时用来计算代数曲线／参数曲线间的最近距离．对于平面二次曲线,采用文中方法得到的单变量多项式方程次数比已有类似方法中结果方程的次数更低,从而可以降低方程求解的计算复杂度或提高求解的稳定性．     

基于几何约束的三次代数曲线插值

尽管三次参数曲线在曲线曲面造型中扮演着主要角色,但是计算几何专家也一直没有放弃对三次代数曲线的性质及应用进行研究。该文首先综述了近年来有关三次代数曲线研究的最新进展,对各主要方法的优缺点进行了客观的评价。然后提出了一种基于几何约束的三次代数曲线的插值方法,该方法守完全通过几何量如控制顶点、切线和曲率来控制三次代数曲线的形状,使得对三次代数曲线的编辑与对三次B－样条曲线的编辑一样灵活方便。该文提出的代数曲线的结构有两种,一种是插值平面上四点及两端点切线的三次代数曲线;另一种是插值两端点、两切线及两曲率的三次代数曲线。在第二种情况下对曲率的情况进行了详细的分类。并且从理论上对曲线的连续性及保凸性进行了严格的证明。     

An offset algorithm for polyline curves

Polyline curves which are composed of line segments and arcs are widely used in engineering applications. In this paper, a novel offset algorithm for polyline curves is proposed. The offset algorithm comprises three steps. Firstly, the offsets of all the segments of polyline curves are calculated. Then all the offsets are trimmed or joined to build polyline curves that are called untrimmed offset curves. Finally, a clipping algorithm is applied to the untrimmed offset curves to yield the final results. The offset algorithm can deal with polyline curves that are self-intersection, overlapping or containing small arcs. The new algorithm has been implemented in a commercial system TiOpenCAD 8.0 and its reliability is verified by a great number of examples.     

Conchoid surfaces of rational ruled surfaces

The conchoid surface G of a given surface F with respect to a point O is roughly speaking the surface obtained by increasing the radius function of F with respect to O by a constant d. This paper studies real rational ruled surfaces in this context and proves that their conchoid surfaces possess real rational parameterizations, independently of the position of O. Thus any rational ruled surface F admits a rational radius function r(u,v) with respect to any point in space. Besides the general skew ruled surfaces and examples of low algebraic degree we study ruled surfaces generated by rational motions.     

Reducibility of offsets to algebraic curves

Computing offset curves and surfaces is a fundamental operation in many technical applications. This paper discusses some issues that are encountered during the process of designing offsets, especially the problems of their reducibility and rationality (which are closely related). This study is crucial especially for formulating subsequent algorithms when the number and quality of offset components must be revealed. We will formulate new algebraic and geometric conditions on reducibility of offsets and demonstrate how they can be applied. In addition, we will present that our investigations can also serve to better understand the varieties fulfilling the Pythagorean conditions (PH curves/PN surfaces). A certain analogy of the PH condition for parameterized curves (or general parameterized hypersurfaces) will be presented also for implicitly given (not necessarily rational) curves (or hypersurfaces).     

一种误差可控的地图边界线的等距线计算方法

在边境地区无线电台站审查、协调需要较高精度地理数据的背景下,提出了一种基于距离场的地图边界线的等距线计算方法。该方法使用规则栅格点构建距离场,通过对误差来源的分析,得到了用栅格点间距表示的误差上界。实际计算时,只需设定一个允许的最大误差值,即可自动完成对指定边界线的计算,得到多边形表示的所需精度的等距线。实验表明基于距离场的方法是可行的并且具有可控的精度,计算的等距线已应用到边境台站的技术审查中。     

An approximation approach of the clothoid curve defined in the interval [0, π/2] and its offset by free-form curves

We propose an algorithm to approximate the clothoid curve defined in the interval [0, π/2] and its offset curves with Bézier curves and the approximation errors converge to zero as the degree of the Bézier curves is increased. Secondly, we discuss how to approximate the clothoid curve by B-spline curves of low degrees. By employing our method, the clothoid curve and its offset can be efficiently incorporated into CAD/CAM systems, which are important for the development of 3D civil engineering CAD systems, especially for 3D highway road design systems. The proposed method has been implemented on AutoCAD R14.     

Automatic parametrization of rational curves and surfaces II: cubics and cubicoids

Algorithms that can obtain rational and special parametric equations for degree three algebraic curves (cubics) and degree three algebraic surfaces (cubicoids), given their implicit equations are described. These algorithms have been implemented on a VAX8600 using VAXIMA.     

The Classical Theory of Invariants and Object Recognition Using Algebraic Curve and Surfaces

Combining implicit polynomials and algebraic invariants for representing and recognizing complicated objects proves to be a powerful technique. In this paper, we explore the findings of the classical theory of invariants for the calculation of algebraic invariants of implicit curves and surfaces, a theory largely disregarded in the computer vision community by a shadow of skepticism. Here, the symbolic method of the classical theory is described, and its results are extended and implemented as an algorithm for computing algebraic invariants of projective, affine, and Euclidean transformations. A list of some affine invariants of 4th degree implicit polynomials generated by the proposed algorithm is presented along with the corresponding symbolic representations, and their use in recognizing objects represented by implicit polynomials is illustrated through experiments. An affine invariant fitting algorithm is also proposed and the performance is studied.     

Surfaces with Pythagorean normals along rational curves

A rational curve on a rational surface such that the unit normal vector field of the surface along this curve is rational will be called a curve providing Pythagorean surface normals (or shortly a PSN curve). These curves represent rational paths on the surface along which the surface possesses rational offset curves. Our aim is to study rational surfaces containing enough PSN curves. The relation with PN surfaces will be also investigated and thoroughly discussed. The algebraic and geometric properties of PSN curves will be described using the theory of double planes. The main motivation for this contribution is to bring the theory of rational offsets of rational surfaces closer to the practical problems appearing in numerical-control machining where the milling cutter does not follow continuously the whole offset surface but only certain chosen trajectories on it. A special attention will be devoted to rational surfaces with pencils of PSN curves.     

Asymptotic behavior of an implicit algebraic plane curve

In this paper, we introduce the notion of infinity branches as well as approaching curves. We present some properties which allow us to obtain an algorithm that compares the behavior of two implicitly defined algebraic plane curves at the infinity. As an important result, we prove that if two plane algebraic curves have the same asymptotic behavior, the Hausdorff distance between them is finite.     

Robust Plotting of Polar Algebraic Curves, Space Algebraic Curves, and Offsets of Planar Algebraic Curves

In this paper, based on modifications to the well-known subdivision scheme in combination with modified affine arithmetic method to guide the subdivision, we propose four robust and reliable algorithms for plotting polar algebraic curves, space algebraic curves and offsets of planar algebraic curves.     

Multiple View Geometry of General Algebraic Curves

We introduce a number of new results in the context of multi-view geometry from general algebraic curves. We start with the recovery of camera geometry from matching curves. We first show how one can compute, without any knowledge on the camera, the homography induced by a single planar curve. Then we continue with the derivation of the extended Kruppa's equations which are responsible for describing the epipolar constraint of two projections of a general algebraic curve. As part of the derivation of those constraints we address the issue of dimension analysis and as a result establish the minimal number of algebraic curves required for a solution of the epipolar geometry as a function of their degree and genus.We then establish new results on the reconstruction of general algebraic curves from multiple views. We address three different representations of curves: (i) the regular point representation in which we show that the reconstruction from two views of a curve of degree d admits two solutions, one of degree d and the other of degree d(d – 1). Moreover using this representation, we address the problem of homography recovery for planar curves, (ii) dual space representation (tangents) for which we derive a lower bound for the number of views necessary for reconstruction as a function of the curve degree and genus, and (iii) a new representation (to computer vision) based on the set of lines meeting the curve which does not require any curve fitting in image space, for which we also derive lower bounds for the number of views necessary for reconstruction as a function of curve degree alone.     

Algebraic pruning: a fast technique for curve and surface intersection

Computing the intersection of parametric and algebraic curves and surfaces is a fundamental problem in computer graphics and geometric modeling. This problem has been extensively studied in the literature and different techniques based on subdivision, interval analysis and algebraic formulation are known. For low degree curves and surfaces algebraic methods are considered to be the fastest, whereas techniques based on subdivision and Bézier clipping perform better for higher degree intersections. In this paper, we introduce a new technique of algebraic pruning based on the algebraic approaches and eigenvalue formulation of the problem. The resulting algorithm corresponds to computing only selected eigenvalues in the domain of intersection. This is based on matrix formulation of the intersection problem, power iterations and geometric properties of Bézier curves and surfaces. The algorithm prunes the domain and converges to the solutions rapidly. It has been applied to intersection of parametric and algebraic curves, ray tracing and curve-surface intersections. The resulting algorithm compares favorably with earlier methods in terms of performance and accuracy.     

代数双曲空间中拟Legendre 基的应用

鉴于Legendre 基等正交基在代数多项式空间中的广泛应用,论文在深入 研究代数双曲空间的拟Legendre 基性质的基础上,给出了其在反函数逼近和等距曲线逼近 上的应用。利用多项式和双曲函数的混合多项式序列来逼近反函数,并通过实例证明给出方 法的有效性;对基曲线的法矢曲线进行逼近,构造H-Bézier 曲线的等距曲线的最佳逼近, 这种方法直接求得逼近曲线的控制顶点,计算简单,截断误差小。     

Application of single-level,pointwise algebraic,and smoothed aggregation multigrid methods to direct numerical simulations of incompressible turbulent flows

Single- and multi-level iterative methods for sparse linear systems are applied to unsteady flow simulations via implementation into a direct numerical simulation solver for incompressible turbulent flows on unstructured meshes. The performance of these solution methods, implemented in the well-established SAMG and ML packages, are quantified in terms of computational speed and memory consumption, with a direct sparse LU solver (SuperLU) used as a reference. The classical test case of unsteady flow over a circular cylinder at low Reynolds numbers is considered, employing a series of increasingly fine anisotropic meshes. As expected, the memory consumption increases dramatically with the considered problem size for the direct solver. Surprisingly, however, the computation times remain reasonable. The speed and memory usage of pointwise algebraic and smoothed aggregation multigrid solvers are found to exhibit near-linear scaling. As an alternative to multi-level solvers, a single-level ILUT-preconditioned GMRES solver with low drop tolerance is also considered. This solver is found to perform sufficiently well only on small meshes. Even then, it is outperformed by pointwise algebraic multigrid on all counts. Finally, the effectiveness of pointwise algebraic multigrid is illustrated by considering a large three-dimensional direct numerical simulation case using a novel parallelization approach on a large distributed memory computing cluster.     

Design of Ada Systems Yielding Reusable Components: An Approach Using Structured Algebraic Specification

Our experience with design of Ada¹ software has indicated that a methodology, based on formal algebra, can be developed which integrates the design and management of reusable components with Ada systems design. The methodology requires the use of a specification language, also based on formal algebra, to extend Ada's expressive power for this purpose. We show that certain requirements for the use of Ada packages which cannot be expressed in Ada can be expressed in algebraic specification languages, and that such specifications can then be implemented in Ada.